Let the logic gates (corresponding to Boolean functions) in a circuit be numbered according to their order of functioning. We introduce the requirement for the output value of a gate (either 0 or 1) to be stored in a register, i.e. a memory element, until all the gates that use this output value have performed their computations. We investigate the Shannon function under the condition that no more than t memory elements are used. We use a set of gates consisting of all p-variable Boolean functions. For t=1 circuits realizing an arbitrary Boolean function may be constructed only if p grows alongside with n. In this case we establish the order of growth of the Shannon function. For a certain range of values of p the Shannon function order of growth is established for t=2. For all fixed t ≥ 3 and p ≥ 2 we establish the asymptotic value of the Shannon function.